7.2 Shear Stress, Shear Flow, and Built-Up Sections

Shear stress is not generally uniform over a beam cross-section. The PE Civil Reference Handbook mechanics formulas distinguish a sectional resultant V, a local stress τ, and an interface force per unit length q. For July 2026, follow the April 2024 Civil: Structural specification, the current handbook, the AISC Steel Construction Manual 15th edition, and NDS 2018 using ASD only for wood. Later reference editions are outside this exam boundary.

Choose the Shear Model

The average expression τ_avg = V/A is useful when a direct shear force is intentionally idealized as uniform across a resisting area, such as a simple pin or punching plane model. It does not reproduce the transverse shear distribution caused by bending in a beam. For an elastic prismatic beam,

τ = VQ/(It)

where:

• V is internal shear at the section along the member;
• I is the second moment of area of the entire transformed or actual section used in the analysis about its neutral axis;
• Q = A'ȳ is the first moment about that neutral axis of the portion A' above or below the evaluation plane; and
• t is the local cross-section width through which shear acts at that plane.

First choose the plane, then identify A' and t. At the neutral axis of a rectangle, A' is the top half and t is the full rectangle width. At a flange-web interface, A' is normally the flange and any material outboard of that interface; t for stress just inside the web is the web thickness. Using the top-half Q at every interface or using flange width as web thickness gives the wrong local stress.

Because Q falls to zero at a free top or bottom surface, transverse shear stress is zero there in the elementary beam model. In a rectangle it varies parabolically and reaches 1.5V/A at the neutral axis. In an I-shape, the thin web carries high shear stress because t is small; a low whole-section average can hide that local demand.

Worked Stress Calculation

A rectangular 4 in by 12 in beam carries internal shear V = 6.00 kips. Find the elastic transverse shear stress at its neutral axis.

For the whole rectangle,

I = bh^3/12 = (4)(12^3)/12 = 576 in^4

The area above the neutral axis is A' = (4)(6) = 24 in^2, and its centroid is 3 in above the neutral axis. Therefore,

Q = A'ȳ = (24)(3) = 72 in^3

At that plane, t = b = 4 in. Thus,

τ_max = VQ/(It)

τ_max = (6,000 lb)(72 in^3)/[(576 in^4)(4 in)] = 187.5 psi

The independent rectangle check gives

1.5V/A = 1.5(6,000)/(4 × 12) = 187.5 psi

so the selected Q, I, and t are consistent. Using V/A = 125 psi would understate the maximum by one-third.

From Shear Stress to Shear Flow

Removing local thickness from the stress equation gives

q = τt = VQ/I

The units are force per unit length: for USCS inputs above, pounds per inch. Shear flow measures the longitudinal force transfer needed between adjacent layers so the cross-section bends together. It is not pressure. For the same 4 in by 12 in section built from two 4 in by 6 in planks joined along the neutral-axis interface,

q = (6,000)(72)/576 = 750 lb/in

If one row of fasteners alone transfers this flow and fasteners are spaced at 3 in, the required force per fastener is

F_fastener = qs = (750 lb/in)(3 in) = 2,250 lb

This demand must be compared with a compatible allowable or design fastener value that accounts for the actual shear planes, spacing, edge distances, material, and load-duration or other code adjustments. If two rows share load, do not divide by two unless symmetry, stiffness, and the problem detail justify equal sharing.

Built-Up and Thin-Walled Details

For a steel plate girder, q at the flange-web junction represents the longitudinal transfer that welds or fasteners must develop. Use the flange-area Q at that junction. For a built-up wood beam, nails, screws, bolts, or adhesive transfer the same type of interface flow, but NDS connection values and ASD-only exam treatment govern capacity. For a multi-cell or open thin-walled shape, direction and continuity of shear flow matter; do not assume one constant value around a changing section without analysis.

A connector group may have more than one interface, row, or shear plane. Sketch the cut surfaces and label which component's longitudinal force changes along the beam. The required transfer over length Δx is approximately qΔx; discrete fasteners replace that continuous transfer at their spacing. At a location where V changes, q changes proportionally, so constant spacing based on maximum shear is conservative only if code detailing requirements are also satisfied.

Exam Workflow and Checks

1. Cut the member at the requested longitudinal position and determine V.
2. Mark the exact cross-section plane where stress or interface transfer is requested.
3. Compute whole-section I about the correct neutral axis.
4. Select the area on one side of the plane and compute its Q.
5. Use local t for stress, or omit t for shear flow.
6. For connectors, multiply q by spacing and distribute only through justified load paths.
7. Keep units explicit: stress is force/area; shear flow is force/length; connector demand is force.

Finally, compare demand with the matching AISC or NDS resistance format. Correct mechanics cannot rescue a capacity check that mixes LRFD demand with ASD resistance, and an average stress cannot replace a required local web, interface, or connector check.